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TOTAL DISPLACEMENT OF LOOPLESS RANDOM WALK

## A for good and evil as well as loopless
## total displacement function of random walk
## that takes in the step numbers as an
## argument and returns the displacement
## of the rnd walker.Note that the probabilities
## to turn rigth and left are equal but the left
## step is twice the rigth step.
## Usage : rw_uneven(N)
function rw_uneven(N)
rn=rand(N,1); ## If the element of a random vector
r=-(rn<0.5); ## is smaller than 0.5, then return -1,
li=find(r == 0); ## Else return 2(equal probability).
r(li)=2;
x=sum(r) ## total displacement
endfunction

## Newton-Rapson Method to the smallest non negative root
## of the 8th degree Legendre Polynomial
## P8(x)=(1/128)(6435x^8-12012x^6+6930x^4-1260x^2+35)
## where -1<=x<=1.
## for the smallest non negative root, we can ignore
## all the terms except the last two by truncated
## the function to be zero and find
## x=0.167 as the initial smallest non negative
## root.
##Constants and initializations
x=[]; ## Empty array for the iterated x roots
x(1)=0.16700000; ## Initial guess to begin the iteration for the
## smallest non-negative root.
L8=[]; ## Empty array for the Legendre polynomial
L8p=[]; ## Empty array for the derivative of the Legendre polynomial
for i=1:100
##The value of the function at x
L8(i)=(1/128)*(6435*x(i)^8-12012*x(i)^6+6930*x(i)^4-1260*x(i)^2+35);
##The value of the derivative of the function at x
L8p(i)=(1/128)*(6435*8*x(i)^7-12012*6*x(i)^5+6930*4*x(i)^3-1260*2*x(i));
x(i+1)=x(i)-L8(i)/L8p(i); ## the iteration
endfor
## For plot let's define a new variable…